Theoretical analysis of genetic algorithms in noisy environments based on a Markov Model

Abstract

In this study, we take a first step towards theoretically analyzing genetic algorithms (GAs) in noisy environments using Markov chain theory. We explicitly construct a Markov chain that models GAs applied to fitness functions perturbed by either additive or multiplicative noise that takes on finitely many values, and we analyze the chain to investigate the transition and convergence properties of the GAs. For the additive case, our analysis shows that GAs eventually (i.e., as the number of iterations goes to infinity) find at least one globally optimal solution with probability 1. In contrast, GAs may eventually with probability 1 fail to do so in the multiplicative case, and we establish a condition that is both necessary and sufficient for eventually finding a globally optimal solution. In addition, our analysis shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property, we derive an upper bound for the number of iterations sufficient to ensure with certain probability that a GA has reached the set of globally optimal solutions and continues to include in each subsequent population at least one globally optimal solution whose observed fitness value is greater than that of any suboptimal solution.